A342817 Power series expansion of AQM(1,1-8x) where AQM denotes the arithmetic-quadratic mean.

1, -4, 4, 16, 52, 112, -48, -1984, -11212, -33360, 6224, 713536, 4441872, 13004480, -17374656, -432012032, -2525831628, -6454496208, 21147389392, 326358047552, 1794285832464, 4124461926592, -19727734694848, -263598020446976, -1416694290412784, -3151402998261312

Offset

0,2

Comments

Generating function: AQM(1,1-8x) where AQM(u,v) (arithmetic-quadratic mean of u and v) is the fixed point obtained by iterating ((u+v)/2, sqrt((u^2+v^2)/2)) (we choose 1-8x in order to avoid denominators, as in A060691).

Links

Table of n, a(n) for n=0..25.

Example

First steps of iteration of ((u+v)/2, sqrt((u^2+v^2)/2)) are (1, 1-8x), (1 - 4*x, 1 - 4*x + 8*x^2 + 32*x^3 + 96*x^4 + O(x^5)), then (1 - 4*x + 4*x^2 + 16*x^3 + 48*x^4 + O(x^5), 1 - 4*x + 4*x^2 + 16*x^3 + 56*x^4 + O(x^5)) and (1 - 4*x + 4*x^2 + 16*x^3 + 52*x^4 + O(x^5), 1 - 4*x + 4*x^2 + 16*x^3 + 52*x^4 + O(x^5)), so the first terms of this sequence are 1, -4, 4, 16, 52.

Prog

(Sage)
R.<x> = PowerSeriesRing(QQ, default_prec=50)
(a, b) = (1, 1-8*x)
for i in range(50):
    (a, b) = ((a+b)/2, sqrt((a^2+b^2)/2))
a.coefficients()
(PARI) seq(n)={my(p=1, q=1-8*x+O(x*x^n)); while(p!=q, my(t=p+q); q = sqrt((p^2 + q^2)/2); p=t/2); Vec(p)} \\ Andrew Howroyd, Mar 22 2021

Crossrefs

Compare A060691 for the arithmetic-geometric mean.

Sequence in context: A223202 A298448 A222144 * A107382 A257622 A185567

Adjacent sequences: A342814 A342815 A342816 * A342818 A342819 A342820

Keyword

sign

Author

David A. Madore, Mar 22 2021

Status

approved